|
Welcome to the sandbox. You can edit this page in any way you want (except please keep the first line intact) to try out editing ideas or to practice using wiki commands before making changes to established articles. Please note that changes made to this page may disappear at any time due to the edits of other users, or as a result of periodic "blanking". Old versions of the page can be recovered by looking in the page history. There is also a template sandbox you can use for testing template ideas. If you have particularly extensive testing to do, you might want to create one or more subpages of your user page — just create a link on your user page of the form [[User:Your username/Subpage title]] (or click here for Sandbox subpage), then follow that "redlink" to edit the page — or simply use your user page itself for trial edits. |
The aim of this seminar is to introduce the theory of Schur functors, with an aim of being able to compute these in specific applications.
Let $K$ be a commutative ring. We recall that the $n$-th tensor power $E^{\otimes n}$ of a module $E$ represents the functor sending a module $M$ to the set (in fact $K$-module) of multilinear maps $E^n\to M$. Similarly we can define the $n$-th exterior power $\bigwedge^n(E)$ as the module representing the functor of alternating multilinear maps, and the $n$-th symmetric power $\mathrm{Sym}^n(E)$ as representing the functor of symmetric multilinear maps. The Schur functors generalise these concepts: for each partition $\lambda$ one obtains a module $S^\lambda(E)$ representing a certain class of multilinear maps such that $S^{(1^n)}(E)\cong\bigwedge^n(E)$ and $S^{(n)}(E)\cong\mathrm{Sym}^n(E)$. We can then regard the map $E\mapsto S^\lambda(E)$ as a functor on the category of $K$-modules, even on the category of finitely-generated free $K$-modules.